Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

Abstract

In the last two decades, the field of geometric numerical integration and structure-preserving algorithms has focused on the design of numerical methods that preserve properties of Hamiltonian systems, evolution problems on manifolds, and problems with highly oscillatory solutions. In this paper, we show that a different geometric property, namely, the blow up of solutions in finite time, can also be taken into account in the numerical integrator, giving rise to geometric methods we call B-methods. We give a first systematic approach for deriving such methods for scalar and systems of semi- and quasi-linear parabolic and hyperbolic partial differential equations. We show both analytically and numerically that B-methods have substantially better approximation properties than standard numerical integrators as the solution approaches the blow-up time.

Keywords

geometric integration, blow-up solutions, nonlinear partial differential equations, nonlinear systems of equations

Publication Date

2015

Source Publication Title

SIAM Journal on Scientific Computing

Volume

37

Issue

6

Start Page

A2298

End Page

A3029

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/15M1011767

Link to Publisher's Edition

https://doi.org/10.1137/15M1011767

ISSN (print)

10648275

ISSN (electronic)

10957197

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